Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394090 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4028 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,13)(8,14)(9,12)(10,15)(11,16)(17,24)(18,32)(19,27)(20,28)(21,25)(22,30)(23,29)(26,34)(31,36)(33,35);(2,4)(6,12)(7,17)(9,10)(14,25)(15,16)(18,31)(19,20)(21,23)(24,27)(29,30)(33,34)(35,36);(5,7)(6,13)(8,18)(9,17)(10,19)(11,20)(12,24)(14,26)(15,27)(16,28)(21,31)(22,34)(23,33)(25,35)(29,36)(30,32);(5,8)(6,14)(7,18)(9,21)(10,23)(11,22)(12,25)(13,26)(15,29)(16,30)(17,31)(19,33)(20,34)(24,35)(27,36)(28,32)Polycyclic presentation:
2^9|e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,g^b=gi,e^c=ei,g^c=gi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72279287, #72312267, #72312291, … |
| maximal normal subgroup of | #72279287, #72312267, #72312291, … |
| maximal quotient of | #72279301, #72279735, #72280517, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301286);